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Fall 1-1-2015

Nonlinear Analysis and Digital Pre-Distortion of the Nonlinear Analysis and Digital Pre-Distortion of the

SC-FDMA Signals in LTE Uplink System SC-FDMA Signals in LTE Uplink System

Changwen Zhu Portland State University

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Recommended Citation Recommended Citation Zhu, Changwen, "Nonlinear Analysis and Digital Pre-Distortion of the SC-FDMA Signals in LTE Uplink System" (2015). Dissertations and Theses. Paper 2105. https://doi.org/10.15760/etd.2103

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Nonlinear Analysis and Digital Pre-Distortion of the SC-FDMA Signals in LTE

Uplink System

by

Changwen Zhu

A thesis submitted in partial fulfillment of the

requirements for the degree of

Master of Science

in

Electrical and Computer Engineering

Thesis Committee:

Fu Li, Chair

Xiaoyu Song

James E. Morris

Portland State University

2014

i

Abstract

Single-carrier frequency division multiple access (SC-FDMA) has become a popular

alternative to orthogonal frequency division multiple access (OFDM) in multi-user

communication on LTE uplink systems. This is primarily due to the low peak-to-

average power ratio (PAPR) of SC-FDMA compared to that of OFDM. Long-term

evolution (LTE) uses SC-FDMA on the uplink to exploit this PAPR advantage to

reduce transmit power amplifier (PA) back-off in mobile user terminals. However, the

latest generation of communication systems requires high power efficiency and a large

quantity of capacity in transmitting mobile data, bringing out some other critical

problems: 1) the nonlinearity of Radio Frequency (RF) power amplifiers inevitably

affects the power efficiency. Working beyond the saturation point is the main reason

for power amplifiers having nonlinear properties; 2) In order to obtain adequate

capacity, wide bandwidth is applied to the latest communication systems. Since in

previous systems the pre-distorter would focus on memory-less distortion with

relatively narrow bandwidth, this change makes memory distortion become a serious

issue, degrading the transmission quality in the wireless communication systems.

The intent of this thesis is to present nonlinear analysis of the SC-FDMA Spectrum

with the RF power amplifier. Relevant mathematical models were considered and

applied to the RF power amplifier in terms of intermodulation products and the third-

order intercept point. The equivalent mathematical model is applied for the first time

to SC-FDMA signals and with the previous formulation of the PA model, the

derivation of the expressions for spectrum regrowth of amplified SC-FDMA signals

ii

was first established and finally simulated with Matlab software. The digital pre-

distortion (DPD) technology was also applied to SC-FDMA signals for the first time

in this thesis. An inverse intermodulation and Autoregressive Moving-Average (IM-

ARMA) model was introduced to linearize the PA distortion with memory in the LTE

uplink system. The DPD was finally implemented by Matlab R2010b.

Conclusions are drawn that amplified power emission levels can be expressed by the

form of third-order intercept point (IP3). The expressions for spectrum regrowth of

amplified SC-FDMA signals have been verified for the first time with the comparison

of simulation, measurement and calculation results. The effects of third order

intermodulation have a greater impact than higher order components with respect to

out-of-band emission power levels. Furthermore, the DPD algorithm reduced the

spectrum regrowth of SC-FDMA signals by 12 dB. The proposed pre-distorter can

effectively solve the distortion problem caused by the memory effect in RF power

amplifier.

iii

Acknowledgements

I feel so fortunate to have all kinds of support from many helpful people. I would like

to express the deepest appreciation to my advisor and thesis committee chair,

Professor Fu Li, for his constant support, encouragement, enthusiasm, and patience in

guiding this research. He introduced me to the various aspects of wireless

communication systems, and helped me to find this exciting research topic. Without

his invaluable probing comments and insightful suggestions this thesis would not have

been possible.

I owe special thanks to Professor Xiaoyu Song and Professors James Morris, for their

guidance and support as the members of the thesis committee. My appreciation should

also be given to Xiao Li, the Ph.D student in our video image processing group, who

provided me the academic quest during the whole time.

My special thanks to my friends in Portland: Ming Shen, Qi Li, Siyuan Yan, Bosi

Chen and Peng Gao. My special thanks also to my friends in the US, UK and China:

Boyang Lu, Linqing Luo, Danjun Xu, Dapeng Yuan, Xuanzi Xia, Yujing Zhang,

Yunwei Wu and Dan Wu.

This thesis is dedicated to my parents Bingguo Zhu and Min Jin, and my grandparents,

for their constant and unconditional support and love. They are my first mentors who

nurtured my growth and instilled in me righteous values for living.

iv

Table of Contents

Abstract ............................................................................................................................ i

Acknowledgements ....................................................................................................... iii

List of Tables ................................................................................................................. vi

List of Figures ............................................................................................................... vii

Chapter 1 Introduction .................................................................................................... 1

1.1 A Statement of the Problem .................................................................................. 1

1.2 Review of Literature ............................................................................................. 2

1.3 Contributions of This Thesis ................................................................................. 2

1.3.1 Nonlinear Analysis of SC-FDMA Spectrum ................................................. 2

1.3.2 The Digital Pre-Distortion of SC-FDMA Spectrum ...................................... 3

Chapter 2 Nonlinear Analysis of SC-FDMA Spectrum for LTE Up-link ...................... 4

2.1 Power Amplifier Model ........................................................................................ 4

2.1.1 Nonlinearity of Power Amplifiers .................................................................. 4

2.1.2 Traditional Memory-less Model .................................................................... 5

2.1.2.1 Intermodulation Products (𝐼𝑀𝑃𝑛) ........................................................... 6

2.1.2.2 Third-order Intercept Point (𝐼𝑃3) ............................................................ 8

2.1.2.3 Fifth-order Intercept Point (𝐼𝑃5) ........................................................... 10

2.1.3 RF Power Amplifier’s Mathematical Model ................................................ 11

2.1.3.1 Mathematical Model of with 𝐼𝑃3 only .................................................. 11

2.1.3.2 Mathematical Model of with 𝐼𝑃3 and 𝐼𝑃5 ............................................ 12

2.2 SC-FDMA Signals .............................................................................................. 12

2.2.1 Basics of SC-FDMA Signals ....................................................................... 13

2.2.2 The Equivalent Mathematical Model of SC-FDMA signals ........................ 14

2.3 Spectrum Regrowth of Amplified LTE Uplink Signals ..................................... 16

Chapter 3 The Digital Pre-Distortion of the SC-FDMA Signals in LTE Up-link

System .......................................................................................................................... 19

3.1 Pre-Distortion Review ........................................................................................ 19

3.2 Spectrum Regrowth Analysis with Pre-Distortion ............................................. 20

v

3.2.1 Block A: The Nonlinear Model .................................................................... 21

3.2.2 Block C: The Memory Effect Model ........................................................... 21

3.2.3 Block B: The Inverse Polynomial of PA ...................................................... 23

3.2.4 Block D: The Inverse Memory Effect of PA ............................................... 23

Chapter 4 Analysis and Experimental Study of SC-FDMA Signals ............................ 25

4.1 SC-FDMA Spectrum for LTE Up-link System .................................................. 25

4.1.1 SC-FDMA Spectrum with 𝐼𝑃3 only ............................................................ 28

4.1.2 SC-FDMA Spectrum with both 𝐼𝑃3 and 𝐼𝑃5 .............................................. 29

4.2 The Digital Pre-Distortion of the SC-FDMA Signals ........................................ 30

Chapter 5 Concluding Remarks .................................................................................... 33

References .................................................................................................................... 34

vi

List of Tables

Table 2.1 Intermodulation distortion products in a Two-Tone System .......................... 7

Table 2.2 Parameters applied in two-tone system .......................................................... 7

Table 2.3 Parameters applied in the simulation math model ........................................ 15

Table 4.1 Experimental Instruments ............................................................................. 27

vii

List of Figures

Figure 2.1 Characteristics of ideal amplifiers ................................................................. 4

Figure 2.2 Characteristics of practical amplifier ............................................................ 5

Figure 2.3 Spectrum of Products in the two-tone system ............................................... 8

Figure 2.4 The definition of third-order intercept point ................................................. 9

Figure 2.5 Two-tone test for the third-order intermodulation ...................................... 10

Figure 2.6 Two-tone test for the fifth-order intermodulation ....................................... 11

Figure 2.7 Transmitter and receiver of OFDM & SC-FDMA systems ........................ 13

Figure 2.8 CCDF of PAPR of OFDM and SC-FDMA signals .................................... 14

Figure 2.9 Power spectrum comparison among simulation, measurement and

calculation results of baseband SC-FDMA signal ........................................................ 16

Figure 3.1 The Structure of DPD process..................................................................... 20

Figure 4.1 The flow chat of simulation process ........................................................... 26

Figure 4.2 Experimental setup of spectrum measurement of SC-FDMA signals ........ 27

Figure 4.3 Power spectrum comparison among amplified signals with 𝐼𝑃3 only in up-

link LTE Wireless system ............................................................................................ 28

Figure 4.4 Power spectrum comparison with both 𝐼𝑃3 and 𝐼𝑃5 .................................. 30

Figure 4.5 The power spectrum comparison of SC-FDMA signals ............................. 31

1

Chapter 1 Introduction

1.1 A Statement of the Problem

The nonlinearity of RF power amplifiers is one of the main concerns in wireless

communication systems. The amplifier won’t have a non-linear problem before it

attains the saturation point. However, as LTE advanced systems require greater

capacity and reliability in transmitting mobile data, the linear region of the amplifier

can’t satisfy the ever-increasing requirement due to the lack of power efficiency. In

this situation, the power amplifier is always designed to operate beyond the saturation

point, which will result in nonlinearity of the amplifier. The nonlinear distortion is one

of the principal reasons for degrading the transmission quality in the wireless

communication systems. SC-FDMA is commonly used in the uplink of LTE and LTE

advanced systems because of the lower PAPR compared with the widely used OFDM

scheme. However, the nonlinearity caused by the characteristic of the RF power

amplifiers is still a critical problem, and there is little research in linearization of SC-

FDMA signals with RF power amplifiers in LTE uplink systems.

Digital pre-distortion (DPD) is generally applied by employing digital baseband

distortion ahead of the amplifier to compensate for the nonlinearity effects and to

provide efficiency enhancements for high power amplifiers in the different

communication systems. The added distortion has totally opposite nonlinear features

from the nonlinearity of the amplifier. With the inclusion of intermodulation products

(IMPs) in the DPD algorithm, the spectrum regrowth of the amplified signals can be

counteracted well in terms of memory-less PA models. The latest generation wireless

2

communication systems, like LTE and LTE advanced systems, require larger

bandwidth, whereas the additional memory effect reduces precision of the traditional

pre-distorter.

1.2 Review of Literature

Previously the research group in Portland State University has already conducted in-

depth studies of the power amplifier nonlinearity and digital pre-distortion. Heng Xiao

built the mathematical relationship between 3rd and 5th order coefficients and intercept

points of RF power amplifiers [1]. Chunming Liu analyzed nonlinearity of RF power

amplifiers with OFDM signals [2]. Bosi Chen extended that knowledge to LTE and

WiMAX systems [3] and Xiao Li established the model of high order spectrum

regrowth of OFDM signals in 4G communication systems and proceeded to pre-

distortion with memory effects [4]. Furthermore, the impact of the power amplifier on

SC-FDMA signals has been discussed in recent studies [5].

1.3 Contributions of This Thesis

1.3.1 Nonlinear Analysis of SC-FDMA Spectrum

In this thesis, the nonlinearity of SC-FDMA signals has been analyzed and predicted

in spectral terms. Owing to the nonlinearity of RF power amplifiers, an analytic model

is introduced on the basis of the third-order intercept point (𝐼𝑃3) since this could be

considered as a weakly nonlinear system [6]. The model is subsequently applied for

3

the first time to SC-FDMA signals for the nonlinear analysis of spectrum regrowth.

Finally the simulation results have been compared with measurement.

1.3.2 The Digital Pre-Distortion of SC-FDMA Spectrum

Digital Pre-Distortion (DPD) based on the efficient and flexible inverse

intermodulation and Autoregressive Moving-Average (IM-ARMA) model is applied

to SC-FDMA signals in the LTE uplink system to linearize the PA distortion with

memory. It captures the nonlinearity effect and the memory effect first with two

separate models based on the Hammerstein structure, and then applies the inverse

polynomial model and the inverse IM-ARMA model to compensate for the two effects

respectively. The improvement of the DPD algorithm for the SC-FDMA signals has

been verified with a series of experimental measurements for the first time in this

thesis.

4

Chapter 2 Nonlinear Analysis of SC-FDMA Spectrum for LTE Up-link

2.1 Power Amplifier Model

2.1.1 Nonlinearity of Power Amplifiers

The nonlinearity of RF power amplifiers is one of the principal reasons for

degradation of the transmission quality in wireless communication systems. Figures

2.1 and 2.2 show that practical amplifiers perform different from the ideal [7]. The

practical amplifier (Figure 2.2) performs as the same as the ideal one (Figure 2.1) only

in the linear region, while in both the cut-off and saturation regions the output suffers

from the nonlinearity of the amplifier [8].

Figure 2.1 Characteristics of ideal amplifiers

5

Figure 2.2 Characteristics of practical amplifier

2.1.2 Traditional Memory-less Model

For an ideal power amplifier, the output performs perfectly linearly, according to

scaling and superposition principles, and is described as

𝑉𝑜𝑢𝑡(𝑡) = 𝐺𝑉𝑖𝑛(𝑡) (2.1)

where output 𝑉𝑜𝑢𝑡(𝑡) is the product of the input 𝑉𝑖𝑛(𝑡) and a scalar coefficient 𝐺

which represents the amplifier gain.

However, for a practical amplifier, as shown in Figure 2.2, the output will saturate to

the maximum output level even if the input continues to rise. Therefore, the Taylor

series [9] is applied in this case

𝑉𝑜𝑢𝑡(𝑡) = 𝐺[𝑉𝑖𝑛(𝑡)] = ∑ 𝐺𝑛𝑉𝑖𝑛𝑛(𝑡)∞

𝑖=0

= 𝐺0 + 𝐺1𝑉𝑖𝑛(𝑡) + 𝐺2𝑉𝑖𝑛2(𝑡) + 𝐺3𝑉𝑖𝑛

3(𝑡) + ⋯ 𝐺𝑛𝑉𝑖𝑛𝑛(𝑡) (2.2)

6

where 𝐺0 is the DC output offset, and 𝐺𝑛 is the coefficient corresponding to the term

𝑉𝑖𝑛𝑛(𝑡). 𝐺1𝑉𝑖𝑛(𝑡) is the first order term which is the linear part similar to the ideal

amplifier. However, the other terms, such as the second order term 𝐺2𝑉𝑖𝑛2(𝑡), cause

nonlinear distortion if not all coefficient 𝐺𝑛 (𝑛 > 1) are zero. Hence, as 𝐺1 must

always be much larger than 𝐺𝑛 (𝑛 > 1), in order to perform as an acceptable linear

amplifier.

2.1.2.1 Intermodulation Products (𝑰𝑴𝑷𝒏)

Intermodulation distortion is widely used to measure the linearity of RF components,

especially power amplifiers [10]. It often occurs when frequencies other than the input

frequencies are generated by the nonlinear features of amplifiers. These are referred to

as intermodulation products called 𝐼𝑀𝑃𝑛 , where ‘n’ represents the order of the

intermodulation products. 𝐼𝑀𝑃𝑛 is a result of an interaction between two signals in an

electronic apparatus.

In a Two-Tone System, the input signal with two frequencies 𝜔1 and 𝜔2 is described

as

𝑉𝑖𝑛(𝑡) = 𝐴1𝑐𝑜𝑠(𝜔1𝑡) + 𝐴2cos (𝜔2𝑡) (2.3)

Intermodulation products at the frequencies indicated in Table 2.1 are close to the

input signals and are difficult to filter out.

7

Frequency

Fifth-order 3𝜔1 − 2𝜔2

Third-order 2𝜔1 − 𝜔2

First Signal 𝜔1

Second Signal 𝜔2

Third-order 2𝜔2 − 𝜔1

Fifth-order 3𝜔2 − 2𝜔1

Table 2.1 Intermodulation distortion products in a Two-Tone System

For a bandpass system, the two frequencies 𝜔1 and 𝜔2 would be relatively similar

(𝜔1 ≈ 𝜔2 ≫ 0). In Figure 2.3, the input frequencies are 0.9985 GHz and 1.0015 GHz

so the spacing between the two adjacent tones is 3 MHz. Additional parameters are

shown in Table 2.2.

Vertical division 10dB

Fundamental -9.14 dBm

Center frequency 1.0 GHz

Span 32.0 MHz

Table 2.2 Parameters applied in two-tone system

The spectrum was measured and recorded in Figure 2.3, it could be concluded that

these third-order terms are always the largest of all intermodulation products. Also

since they are close to the original signals and hard to filter, they are always

8

considered as the principal concern for filtering the interference out of the desired

signal.

Figure 2.3 Spectrum of Products in the two-tone system

2.1.2.2 Third-order Intercept Point (𝑰𝑷𝟑)

The third-order intercept point (𝐼𝑃3) is an imaginary point in theory, where the third-

order intermodulation distortion signal has the same amplitude as the output signals,

shown in Figure 2.4 [8]. The third-order intercept point is regarded as an effective

measuring method for weakly nonlinear systems. With applying a low-order

polynomial, inferred from the Taylor series expansion, the third-order intercept point

helps build a meaningful model of linear amplified signals including third-order

nonlinear products.

Pt (dB)

Frequency (f)

Pt (Fundamental)

Third-order

Fifth-order

Third-order

Fifth-order

9

Figure 2.4 The definition of third-order intercept point

A higher power output at the intercept point corresponds to better linearity and lower

IMD [8]. Hence the value of 𝐼𝑃3 becomes important when evaluating the operational

range of the amplifier, which relates to the maximum signal power without IMD

occurring. To measure the level of co-channel interference using intercept points, a

two-tone test is applied to measure the model expression of 𝐼𝑃3, which is written as [1]

𝐼𝑃3 = 𝑃𝑡 +𝐼𝑀3

2 (2.4)

where 𝑃𝑡 is the power of the original output tone signal and 𝐼𝑀3 is the difference

between 𝑃𝑡 and the 3rd order intermodulation output power, shown in Figure 2.5 [2].

10

3IM3IMP

Frequency ( f )

dBPttP tP

3IMP

Third-order

IM Product

Figure 2.5 Two-tone test for the third-order intermodulation

2.1.2.3 Fifth-order Intercept Point (𝑰𝑷𝟓)

The fifth-order Intercept Point (𝐼𝑃5) is defined as the point where the fundamental and

𝐼𝑀5 response intersect [11]. The nonlinearity of the power amplifier is always

described and determined by 𝐼𝑃3 provided by the manufacturer’s datasheets. However,

the spectrum could gain increased precision by including both 𝐼𝑃3 and 𝐼𝑃5. Since 𝐼𝑃5

is not given by the manufacturer’s datasheets, a two-tone test is used, as for 𝐼𝑃3, to

measure the model expression of 𝐼𝑃5, which is written as [2]

𝐼𝑃5 = 𝑃𝑡 +𝐼𝑀5

4 (2.5)

where 𝐼𝑀5 is the difference between 𝑃𝑡 and the 5th order intermodulation output power,

shown in Figure 2.6 [2].

11

5IMP5IMP 5IM3IMP

Frequency ( f )

dBPt

tPtP

3IMP

Fifth-order

IM Product

Figure 2.6 Two-tone test for the fifth-order intermodulation

2.1.3 RF Power Amplifier’s Mathematical Model

2.1.3.1 Mathematical Model of with 𝑰𝑷𝟑 only

Owing to the nonlinearity of RF power amplifiers, the Taylor model is established on

the basis of the intercept points (𝐼𝑃𝑛), and is fitted for weakly nonlinear systems and

devices to:

𝑦(𝑡) = ∑ 𝑎2i+1𝑠2𝑖+1(𝑡)∞𝑖=0 (2.6)

where 𝑦(𝑡) is the output and s(𝑡) is input signal of the power amplifier in the

passband [12]. Because even-order terms, which are more than one carrier frequency

away from the centre frequency, can be easily filtered out[1], only the third order

intercept point is considered for the analytical model in this section, expressed as

y(𝑡) = 𝑎1s(𝑡) + 𝑎3s3(𝑡) (2.7)

12

The coefficients 𝑎1 and 𝑎3 depend on the linear gain 𝐺 of the amplifier and 𝐼𝑃3, both

of which are always introduced in datasheets from the manufacturers of RF power

amplifiers. For the amplifier with gain compression (𝑎3 < 0),

𝑎1 = 10𝐺

20, 𝑎3 = −2

310(−

𝐼𝑃310

+3𝐺

20) (2.8)

2.1.3.2 Mathematical Model of with 𝑰𝑷𝟑 and 𝑰𝑷𝟓

In this section, both 𝐼𝑃3 and 𝐼𝑃5 are considered for the analytical model, expressed as

y(𝑡) = 𝑎1s(𝑡) + 𝑎3s3(𝑡) + 𝑎5s5(𝑡) (2.9)

The extra coefficient 𝑎5 depends on the linear gain 𝐺 and 𝐼𝑃5. For the amplifier with

gain compression,

𝑎5 = −2

510(−

𝐼𝑃55

+𝐺

4) (2.10)

2.2 SC-FDMA Signals

Single-Carrier Frequency-Division Multiple Access (SC-FDMA) is widely used in the

up-link of long-term evolution (LTE) and LTE-Advanced systems. It is similar to

Orthogonal Frequency-Division Multiplexing (OFDM) used in the downlink of LTE

systems [13], but with an extra Discrete Fourier Transform (DFT) block prior to

subcarrier mapping, SC-FDMA shows its strength in a robust high peak-to-average-

power ratio (PAPR) which produces benefits of both terminal expense and power

efficiency [14]. However, the nonlinearity caused by the characteristic of the RF

power amplifiers is still a critical problem, and there is very little research into

linearization for the SC-FDMA signals.

13

2.2.1 Basics of SC-FDMA Signals

SC-FDMA with cyclic prefix (CP) is widely applied in the up-link of long-term

evolution (LTE) systems due to its salient advantage of low peak-to-average-power

ratio (PAPR) contributing to a low terminal expense and high power efficiency of the

transmitter [14]. It has some similarities to Orthogonal Frequency-Division

Multiplexing (OFDM) used in the down-link of LTE systems, but the unique

characteristic of SC-FDMA is embodied in different carrier transmission schemes,

prompting SC-FDMA to be a more valuable alternative to OFDM. A SC-FDMA

transceiver occupies an additional Discrete Fourier Transform (DFT) block prior to

subcarrier mapping, illustrated in Figure 2.7 [15].

Figure 2.7 Transmitter and receiver of OFDM & SC-FDMA systems

In a transmitter, data symbols are transformed from the time domain to the frequency

domain. After subcarrier mapping, the Inverse Discrete Fourier Transform (IDFT)

operation performs the same feature with a single carrier signal with lower power

variations. Moreover, the transmitted signals are operated in the frequency domain

through shifting the IDFT inputs, in order to satisfy requests for flexible bandwidth

[16]. By contrast, because of the increasing number of subcarriers, outputs in the time

14

domain for OFDM signals behave like Gaussian noise performing high PAPR values

compared with SC-FDMA signals in Figure 2.8 [17]. On the vertical axis, the value

represents the probability that the PAPR value is larger than the PAPR0 value.

Figure 2.8 CCDF of PAPR of OFDM and SC-FDMA signals

2.2.2 The Equivalent Mathematical Model of SC-FDMA signals

The nonlinearity of SC-FDMA signals is analysed and predicted with respect to the

spectrum.

The analytical model of a SC-FDMA symbol is expressed as [18]

𝑏(𝑡) = 𝑤𝑇(𝑡) ∑ 𝑎𝑘,𝑛𝑁𝑆𝑇/2𝑘=−𝑁𝑆𝑇/2 𝑒𝑗2𝜋𝑘𝛥𝑓(𝑡−𝑇𝐺𝐼) (2.11)

A SC-FDMA symbol consists of a set of modulated data named as 𝑎𝑘,𝑛 in expression

(2.11) showing the data in the nth SC-FDMA symbol and the kth subcarrier. In the SC-

FDMA system, a time-window function 𝑤𝑇(𝑡) is utilized with a rectangular shape.

NST is the total number of subcarriers. 𝛥𝑓 is the sub-carrier spacing while 𝑇𝐺𝐼 means

the guard interval time for creating the cyclic prefix. 𝑟(𝑡) is the concatenation of these

SC-FDMA symbols, which is described as [18]

15

𝑟(𝑡) = ∑ 𝑤𝑇(𝑡 − 𝑛𝑇𝑠)∞𝑛=−∞ [∑ 𝑎𝑘,𝑛

𝑁𝑆𝑇/2

𝑘=−𝑁𝑆𝑇

2

𝑒𝑗2𝜋𝑘𝛥𝑓(𝑡−𝑇𝐺𝐼−𝑛𝑇𝑠)] (2.12)

where 𝑇𝑠 is the symbol duration time. The transmitted SC-FDMA signal 𝑠(𝑡) then is

written as [13]

𝑠(𝑡) = 𝑅𝑒[𝑟(𝑡)𝑒𝑗2𝜋𝑓𝑐𝑡] (2.13)

where 𝑅𝑒[∗] is the real part of the signal while 𝑓𝑐 denotes the carrier frequency of the

LTE system. The parameters used in the math model for testing the LTE signal are

listed in Table 2.3 [18].

SC-FDMA parameter LTE

Subcarrier frequency spacing 𝛥𝑓 15k Hz

Number of used subcarriers 𝑁𝑆𝑇 1200

FFT size 8192

Total symbol time 𝑇𝑠 71.36 μs

Data modulation 16-QAM

Sampling frequency 1/𝑇𝑠 15.36 MHz

Circular prefix 𝑇𝐺𝐼 4.69 μs

Table 2.3 Parameters applied in the simulation math model

The general formula of the power spectral density (PSD) of a SC-FDMA baseband

signal is written as [4]

P𝑟(𝑓) =𝑅𝑠

4∑ [|𝑊(𝑓 − 𝑘∆𝑓)|2 + |𝑊(−𝑓 − 𝑘∆𝑓)|2]NST/2

k=−NST

2

(2.14)

16

where 𝑅𝑠 =1

𝑇𝑠 is the symbol rate and 𝑊(𝑓) defines the Fourier transform of the

rectangular window function in this case. Figure 2.9 illustrates that the simulated and

measured baseband signal spectra achieve good matches with the spectrum of the

calculated expression (2.14) [19].

Figure 2.9 Power spectrum comparison among simulation, measurement and

calculation results of baseband SC-FDMA signal

2.3 Spectrum Regrowth of Amplified LTE Uplink Signals

From expression (2.13), a RF power amplifier has the analytical model shown as

𝑦(𝑡) = 𝛰{𝑠(𝑡)} = Ϝ[𝑟(𝑡)]𝑐𝑜𝑠 (2𝜋𝑓𝑐𝑡 + 𝜑[𝑟(𝑡)]) (2.15)

where 𝛰{∗} is the operation of amplifier while Ϝ[∗] and 𝜑[∗] are AM/AM and

AM/PM conversions respectively, depending on the model characteristics of the

amplifier.

17

Implementing an autocorrelation of 𝑟(𝑡) and by the corollary to the Wiener-

Khintchine Theorem [12], the final power spectrum 𝑃𝑦(𝑓) of y(t) will become:

𝑃𝑦(𝑓) = 𝐴1𝑃1(𝑓−𝑓𝑐)

𝑁𝑆𝑇+ 𝐴2

𝑃3(𝑓−𝑓𝑐)

𝑁𝑆𝑇3 (2.16)

𝑃1 represents the PSD of a SC-FDMA baseband signal and 𝑃3 is equivalent to the

convolution results of three 𝑃1 values. 𝐴1 and A2 are two factors having values after

calculation shown below:

𝐴1 = 2𝑃0 − 12𝑃0210−

𝐼𝑃310 + 18𝑃0

310−𝐼𝑃35

𝐴2 = 48𝑃0310−

𝐼𝑃35 (2.17)

where 𝑃0 =𝑎1

2𝑁𝑆𝑇

4 is the linear output power of the amplifier.

Another form in terms of 𝐼𝑃3, evolved from the power spectrum result of 𝑃𝑦(𝑓) in

(2.16) and (2.17), is described as:

𝐷1 × 10−𝐼𝑃3

5 + 𝐷2 × 10−𝐼𝑃310 + 𝐷𝐶 = 0 (2.18)

where

𝐷1 =18𝑃0

3

𝑁𝑆𝑇3 𝑃1(𝑓 − 𝑓𝑐) +

48𝑃03

𝑁𝑆𝑇3 𝑃3(𝑓 − 𝑓𝑐)

𝐷2 = −12𝑃0

2

𝑁𝑆𝑇3 𝑃1(𝑓 − 𝑓𝑐) (2.19)

𝐷𝐶 =2𝑃0

𝑁𝑆𝑇 𝑃1(𝑓 − 𝑓𝑐) − 𝑃𝑦(𝑓)

In order to get the explicit expression of 𝐼𝑃3, the form in (2.18) is solved with (2.19):

𝐼𝑃3 = −10 × log10 (−𝐷2+√𝐷2

2−4𝐷1𝐷𝐶

2𝐷1) (2.20)

18

According to the theoretical expression, the value of 𝐼𝑃3 depends on a particular out-

of-band emission power level. The emission level of the power spectrum in a given

band from 𝑓1 and 𝑓2 is described as 𝑃𝐼𝑀3(𝑓1, 𝑓2) [13]:

𝑃𝐼𝑀3(𝑓1, 𝑓2) = ∫ 𝑃𝑦(𝑓)𝑑𝑓

𝑓2

𝑓1 (2.21)

From the integration of 𝑃𝑦(𝑓) , another group of coefficients 𝐷1′ , 𝐷2

′ and 𝐷𝐶′ is

calculated as [13]

𝐷1′ =

18𝑃03

𝑁𝑆𝑇3 ∫ 𝑃1(𝑓 − 𝑓𝑐)𝑑𝑓

𝑓2

𝑓1

+48𝑃0

3

𝑁𝑆𝑇3 ∫ 𝑃3(𝑓 − 𝑓𝑐)𝑑𝑓

𝑓2

𝑓1

𝐷2′ = −

12𝑃02

𝑁𝑆𝑇3 ∫ 𝑃1(𝑓 − 𝑓𝑐)𝑑𝑓

𝑓2

𝑓1 (2.22)

𝐷𝐶′ =

2𝑃0

𝑁𝑆𝑇 ∫ 𝑃1(𝑓 − 𝑓𝑐)𝑑𝑓

𝑓2

𝑓1

− 𝑃𝐼𝑀3(𝑓1, 𝑓2)

However, the new expression of 𝐼𝑃3 is of the same form with (2.20) except the latter

coefficients substitute for the previous ones respectively, giving a clear view of state

transition between the out-of-band emission power level and 𝐼𝑃3.

19

Chapter 3 The Digital Pre-Distortion of the SC-FDMA Signals in LTE Up-link

System

RF power amplifiers (PA) bring nonlinear distortion that degrades signal transmission

quality and increases inter-channel interference [20]. In addition, with the

development of fourth generation (4G) mobile communication systems, the modern

wideband signal has aroused the attention to the issues wherein the accuracy of PA

modelling has been influenced by memory effects, which is always not considered in

other narrowband applications [21]. This new issue arises a request for applying DPD

algorithms with the memory effect to the wideband LTE systems since traditional pre-

distorters always perform poor accuracy with memory-less models. A DPD algorithm

is applied to the SC-FDMA signals in the uplink of the LTE systems. It captures the

nonlinearity and memory effects with two separate models based on the Hammerstein

structure first, and then applies the inverse polynomial model and the inverse IM-

ARMA model to compensate previous two parts of effect respectively. The

improvement of the DPD algorithm for the SC-FDMA signals was verified with a

series of experimental measurements.

3.1 Pre-Distortion Review

The digital pre-distorter (DPD) is widely applied by employing digital baseband

distortion ahead of the amplifier to compensate for the nonlinearity effects and provide

efficiency enhancements for high power amplifiers in the different communication

systems. This thesis presents a digital pre-distorter first applied to linearize SC-FDMA

20

signals having the PA distortion with memory effect using an efficient and flexible

inverse intermodulation and Autoregressive Moving-Average (IM-ARMA) model in

the LTE uplink system.

3.2 Spectrum Regrowth Analysis with Pre-Distortion

The DPD algorithm is divided into two parts shown in Figure 3.1 [4]:

Figure 3.1 The Structure of DPD process

The PA is modelled by two blocks: a memory-less nonlinear block (A) and a LTI

(linear time invariant) block (C) presenting memory effect; The digital pre-distorter is

modelled by two inverse functions according to PA model: inverse nonlinear block (B)

and an inverse LTI block (D). The inverse nonlinear block (B) compensates the

memory-less distortion of the PA modelled by block (A), and the inverse LTI block (D)

cancels the memory effect presented by block (C) [4]. The description of each block

will be discussed below.

21

3.2.1 Block A: The Nonlinear Model

The Taylor series model v(n) of a high power amplifier is applied for the polynomial

model, expressed as [22]:

v(n) = 𝑎1w(𝑛) + 𝑎3w3(𝑛) + ⋯ + 𝑎𝑀𝑤𝑀(𝑛) (3.1)

where w(𝑛) represents the input. The coefficient series 𝑎𝑚 is determined by the linear

gain 𝐺 of the amplifier and 𝑚-order intercept point (𝐼𝑃𝑚). It should be noticed that

only terms with odd-order features are included in this model due to the simple

filtration process with even-order 𝐼𝑃𝑚 terms, which are always kept far away from the

center of passband. Moreover, the coefficient series 𝑎𝑚 have the form of an

exponential function. Accordingly, following function is proposed to describe the

relationship between 𝑎𝑚 and PA intercept points [4]:

𝑎𝑚 = −2

𝑚−12

(𝑚

𝑚+12

)

10(𝐼𝑀𝑃𝑚−𝑚𝑃𝑡+𝑚𝐺

20) (3.2)

where (𝑚

𝑚+1

2

) gives the number of different combinations of 𝑚+1

2 elements that can be

chosen from an 𝑚-element set. 𝐺 is the gain of the PA while 𝐼𝑀𝑃𝑚 stands for the 𝑚th

order intermodulation product, which is the difference between the original output

signal and the 𝑚th order intermodulation output power from a simple two tone test.

3.2.2 Block C: The Memory Effect Model

With the development of fourth generation mobile communication system, the modern

wideband signal has aroused the attention to the issues wherein the accuracy of PA

22

modelling has been influenced by memory effects which is always not considered in

narrowband applications.

For the memory effect part, the output is determined by both current and past inputs

[4]. Therefore, the time delays of both inputs and outputs are required to be involved

in the algorithm. Due to the linear delay property of the memory effect, the

Autoregressive Moving-Average (ARMA) model is used as:

∑ 𝑝(𝑘)𝑦(𝑛 − 𝑘) = ∑ 𝑞(𝑙)𝑣(𝑛 − 𝑙)𝐿𝑙=0

𝐾𝑘=0 (3.3)

where 𝑣 and 𝑦 mean the input and output functions, while 𝑘 and 𝑙 are two delays of

outputs and inputs respectively. p(k) and q(l) are coefficients of AR and MA parts of

the model. The transfer function is presented as [4]:

𝐻(𝑧) =∑ 𝑞(𝑘)𝑧−𝑘𝐾

𝑘=0

∑ 𝑝(𝑙)𝑧−𝑙𝐿𝑙=0

(3.4)

The advantages of the ARMA filter contain: It is quite simple to obtain the

coefficients from several mature algorithms [22], also the memory effect can be

reduced easily by applying the pre-distortion algorithm which is the inverse transfer

function of the ARMA filter.

Finally the PA model with memory effect is described with nonlinearity polynomial

function and memory effect function [4]:

∑ 𝑞(𝑘)𝑦(𝑛 − 𝑘)𝐾𝑘=0 = ∑ 𝑝(𝑙)[𝑎1𝑤(𝑛 − 𝑙) + 𝑎3𝑤3(𝑛 − 𝑙) + ⋯ + 𝑎𝑀𝑤𝑀(𝑛 − 𝑙)]𝐿

𝑙=0

(3.5)

23

3.2.3 Block B: The Inverse Polynomial of PA

To pre-distort the PA, we start from the linearization of the memory-less distortion

modelled in (3.1) for the block A. A developed formula based on the polynomial

model is described as [4]:

𝑤(𝑡) = 𝑏1𝑢(𝑡) + 𝑏3𝑢3(𝑡) + ⋯ + 𝑏𝑃𝑢𝑃(𝑡) (3.6)

By replacing 𝑤(𝑡) in (3.6), the whole system in (3.1) then can be expressed as:

𝑣(𝑡) = ℎ1𝑢(𝑡) + ℎ3𝑢3(𝑡) + ⋯ + ℎ𝑀∙𝑃𝑢𝑀∙𝑃(𝑡) (3.7)

The 𝑝th order nonlinearity requests at least the same order polynomial pre-distorter for

linearization. From the definition of 𝑝th order inverse method [22], the coefficients are

described as:

ℎ1 = 𝑎1, ℎ3 = 0, … ℎ𝑝 = 0 (3.8)

PA will have no nonlinearity output with 𝑝th of pre-distorter polynomial order if all the

equations in (3.8) are satisfied. To achieve the goal, the series of coefficient 𝑏𝑃 is

calculated as [4]:

𝑏𝑝 = − ∑ [(∑ 𝑏𝑖1𝑏𝑖2

… 𝑏𝑖𝑚

∑ 𝑖𝑘𝑚𝑘=1 =𝑝

∑ 𝑖𝑘𝑚𝑘=1 =𝑚

𝑖𝑘 𝑖𝑠 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

)𝑎𝑚

𝑎1]𝑀

𝑚=3𝑚 𝑖𝑠 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

(3.9)

where M is the maximum order within the PA model.

3.2.4 Block D: The Inverse Memory Effect of PA

Since linear signals v(t) has been generated by counteracting the effect from both the

inverse polynomial block (B) and the polynomial block (A), the memory effect is

necessary to be removed then. The memory effect (LTI block C) is modelled as

24

ARMA in (3.5), so the inverse block (D) should have the same ARMA structure with

inverse AR and MA coefficients. This was achieved by exchanging the positions of

pole and zero. The output u(n) of inverse ARMA block (D) and the inverse transfer

function can be indicated as:

∑ 𝑝(𝑘)𝑦(𝑛 − 𝑘) = ∑ 𝑞(𝑙)𝑣(𝑛 − 𝑙)𝐿𝑙=0

𝐾𝑘=0 (3.10)

𝐻(𝑧) =∑ 𝑞(𝑘)𝑧−𝑘𝐾

𝑘=0

∑ 𝑝(𝑙)𝑧−𝑙𝐿𝑙=0

(3.11)

25

Chapter 4 Analysis and Experimental Study of SC-FDMA Signals

4.1 SC-FDMA Spectrum for LTE Up-link System

The simulation section followed a similar sequence to the transmitter structure in

Figure 2.7. The simulation process is shown in Figure 4.1. 16-QAM modulated

complex symbols were applied to the SC-FDMA system. The normalization factor

KMOD was chosen as 1

√10 to achieve the same average power for mappings in this

case. Compared with the OFDM system, the extra N-point DFT block transferred the

system into a single-carrier one which avoids a high PAPR and improves system

throughput by confining the power within a certain threshold. Then during the

subcarrier mapping, the Localized FDMA method was chosen through using a certain

fraction of the whole bandwidth. The amplifier simulation was then built with the

Taylor model. For the calculation section, a rectangular window function for the SC-

FDMA signal was applied prior to calculating the PSD of the RF signal. Then the total

bandwidth was split in two sets of parts shown in expression (2.13) and (2.14), and the

whole analytical models were shown and discussed in previous sections.

26

Figure 4.1 The flow chat of simulation process

Finally, the measurement was made with the instruments in Figure 4.2. The Model

types and descriptions are included in Table 4.1. The LTE signal was generated by

Agilent E4438C ESG vector signal generator using a carrier frequency of 1 GHz, and

power level set as -10 dBm. The sampling frequency was 122.88 MHz with 801 points

acquired and analyzed for outputs. The analyzing span was set as 100.0 MHz and

27

reference level as -2.00 dBm. The ZFL-1000LN RF power amplifier was placed

between the generator and the Tektronix RSA 6120A real-time spectrum analyser. It

had a 13 dBm 𝐼𝑃3 and an output gain of 20dB according to the datasheet. The output

spectrum of the amplified SC-FDMA signal from the spectrum analyzer was then

reconstructed by Matlab R2012b.

Figure 4.2 Experimental setup of spectrum measurement of SC-FDMA signals

Description Model No.

Agilent Signal Generator E4438C

Mini-Circuits RF Power Amplifier ZFL-1000LN+

Heathkit Power Supply IP 2718

Tektronix Spectrum Analyzer RSA6120A

Table 4.1 Experimental Instruments

28

4.1.1 SC-FDMA Spectrum with 𝑰𝑷𝟑 only

Figure 4.3 shows that the final simulated and measured RF amplifier spectra have high

similarity to the calculated spectrum from (2.15) in both the pass-band and the

shoulder area.

Figure 4.3 Power spectrum comparison among amplified signals with 𝐼𝑃3 only in up-

link LTE Wireless system

From Figure 4.3, the bandwidth B is observed as about 10 MHz. For |𝑓|< B, where 𝑓

equals the frequency in the horizontal axis in the figure, this part is known as the pass-

band containing the linear output power density for an ideal power amplifier

performance. For the out-of band range, with the frequency range approximately from

29

10 to 30 MHz shown in Figure 4.3, the PSD output 𝑃𝑦(𝑓) is mainly related to 3rd

intermodulation since only 𝐼𝑃3 is considered in the calculation. The result shows that

the out-of-band spectrum density is always controlled by intermodulation, and these

out-of-band frequency components are usually named as spectrum regrowth and

include the most significant part of the emission of a power amplifier.

Furthermore, for the band |𝑓| > 3B, which is about 30 MHz shown in Figure 4.3, the

power density is still not zero because there always exists some noise, which cannot be

neglected in terms of power level compared with the output spectrum. This point of

view is not proposed in Figure 4.3 as the result of some limitations. Firstly, only 𝐼𝑃3 is

considered in this part. From the figure, it can be concluded that it has great difficulty

distinguishing 𝐼𝑃3 with ambient noise since the power level of higher-order

intermodulation points is not considerable in this part, which is more than 40 dB lower

than the amplified output power. Secondly, because the datasheet of a power amplifier

does not often include the value of 𝐼𝑃5 or higher orders, it is not reliable to simulate

and calculate the amplified spectrum with higher order intermodulation at this point.

4.1.2 SC-FDMA Spectrum with both 𝑰𝑷𝟑 and 𝑰𝑷𝟓

From the Figure 4.4, the bandwidth B is modified to about 4.5 MHz. In this case, The

PSD outputs of simulated and measured signal spectrums achieve good matching with

the spectrum of calculated expression for the band |𝑓| < 3B. For the band |𝑓| > 3B,

which is about 13.5 MHz shown in Figure 4.4, The extra noise has a relatively deep

influence on the output spectrum with fifth order intercept points in terms of power

30

level. Furthermore, as referred before, the value of 𝐼𝑃5 is measured and calculated by

two-tone test with (2.5), so some errors may be generated since the amplifier is not

ideal.

Figure 4.4 Power spectrum comparison with both 𝐼𝑃3 and 𝐼𝑃5

4.2 The Digital Pre-Distortion of the SC-FDMA Signals

The same setup of instruments shown in Figure 4.2 and Table 4.1 was applied to

verification of the DPD algorithm. The SC-FDMA signal was generated by the

Agilent E4438C ESG vector signal generator using a carrier frequency of 1 GHz, and

power level -17 dBm. The bandwidth of the signals is 10 MHz. A Tektronix RSA

6120A real-time spectrum analyser was used to measure the original and the amplified

SC-FDMA signals through the ZFL-1000LN RF power amplifier produced by Mini –

31

Circuit. The output gain is 20 dB according to the datasheet. The DPD is finally

implemented by MATLAB R2010b.

Figure 4.5 The power spectrum comparison of SC-FDMA signals

Figure 4.5 shows the power spectrum of the original input signal without amplification,

the amplified signal without DPD, and the amplified signal with DPD. The order of

the nonlinear model we used is up to 5 and the memory depth is 2 for the AR part and

2 for the MA part in the block C and block D. With the DPD algorithm, the amplified

power spectrum is very similar to the original input spectrum after the normalization.

The PAPR (Adjacent channel power ratio) is improved with 12 dB achieved in the

area close to the fundamental frequency. The performance in the further area (offset >

|12 dB|) is a little worse because the inverse polynomial could force the signals to have

no nonlinearity up to the pth order of the pre-distorter polynomial, regardless of the

PA’s polynomial order. Certain nonlinear terms of orders higher than p may still exist

32

in the compensated output. However, those interferences could be filtered out easily

since they are relatively far from the fundamental frequency.

33

Chapter 5 Concluding Remarks

The research results confirm that spectrum regrowth with RF power amplifiers has

been specified and measured efficiently. This sort of design includes the advantages of

SC-FDMA and is of great help for both designing RF power amplifiers and for further

improving spectrum regrowth with pre-distortion outside channels, which can be

extended to other wireless systems based on SC-FDMA modulation. The spectrum

regrowth analysis technology of SC-FDMA signals is applied and verified in the

testing and simulation work with LTE signals. From the experimental result, it was

supposed that the fifth or higher order intermodulation has little effect on out-of-band

emission power levels. The theoretical result from reasonable calculation is verified

along with experiment outputs. The relationship between power spectrum levels and

nonlinearity coefficients of RF power amplifiers, mainly 𝐼𝑃3 in this thesis, is verified

in the analytical model. Also the ARMA based DPD algorithm with memory

polynomial model is applied to SC-FDMA signals in the LTE uplink system. The

experimental results showed that the DPD algorithm reduced the spectrum regrowth

by 12 dB. With the expression of amplified power emission levels we derived, the

result may help meet the out-of-band requirements of the LTE standards and make

progress with digital pre-distortion in the up-link of LTE systems.

34

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